arXiv Math @arxiv_math@qoto.org

The author proves that there are infinitely many primes $p$ such that $\| αp - β\| < p^{-\frac{28}{87}}$, where $α$ is an irrational number and $β$ is a real number. This sharpens a result of Jia (2000) and provides a new triple $(γ, θ, ν)=(\frac{59}{87}, \frac{28}{87}, \frac{1}{29})$ that can produce primes in Ford and Maynard's work on prime-producing sieves. Our minimum amount of Type-II information required ($ν= \frac{1}{29}$) is less than any previous work on this topic using only traditional Type-I and Type-II information.

We characterize intermediate $\mathbb{R}$-algebras $A$ between the ring of semialgebraic functions ${\mathcal S}(X)$ and the ring ${\mathcal S}^*(X)$ of bounded semialgebraic functions on a semialgebraic set $X$ as rings of fractions of ${\mathcal S}(X)$. This allows us to compute the Krull dimension of $A$, the transcendence degree over $\mathbb{R}$ of the residue fields of $A$ and to obtain a Łojasiewicz inequality and a Nullstellensatz for archimedean $\mathbb{R}$-algebras $A$. In addition we study intermediate $\mathbb{R}$-algebras generated by proper ideals and we prove an extension theorem for functions in such $\mathbb{R}$-algebras.

It is proved that for any non-empty finite subset $Q$ of the square numbers, $ |Q+Q|\geq C'|Q|(\log |Q|)^{1/3+o(1)} $.

In this note, we give a characterization of immersed submanifolds of simply-connected space forms for which the quotient of the extrinsic diameter by the focal radius achieves the minimum possible value of $2$. They are essentially round spheres, or the ``Veronese'' embeddings of projective spaces. The proof combines the classification of submanifolds with planar geodesics due to K. Sakamoto with a version of A. Schur's Bow Lemma for space curves. Open problems and the relation to recent work by M. Gromov and A. Petrunin are discussed.

In this study, we address the challenges associated with accurately determining gaze location on a screen, which is often compromised by noise from factors such as eye tracker limitations, calibration drift, ambient lighting changes, and eye blinks. We propose the use of an extended Kalman filter (EKF) to smooth the gaze data collected during eye-tracking experiments, and systematically explore the interaction of different system parameters. Our results demonstrate that the EKF significantly reduces noise, leading to a marked improvement in tracking accuracy. Furthermore, we show that our proposed stochastic nonlinear dynamical model aligns well with real experimental data and holds promise for applications in related fields.

We present a comprehensive study on $SIM(2)$ and $ISIM(2)$ groups, their representations and algebraic aspects. After obtaining $SIM(2)$ through the Inönü-Wigner contraction procedure, a complete four-dimensional algebraic representation is shown for $\mathfrak{sim(2)}$ and $\mathfrak{isim(2)}$. Besides that, we apply Bargmann's formalism to investigate the (projective) representations for both cases, keeping track of the source of phase factors. We complete the study by presenting a particularly simple analysis to probe the existence of local phase factors, which is useful when dealing with non-abelian groups.

Let $G$ be an infinite countable amenable group and let $(X,G)$ be a $G$-subshift with specification, containing a free element. We prove that $(X,G)$ is universal, i.e., has positive topological entropy and for any free ergodic $G$-action on a standard probability space, $(Y,ν,G)$, with $h(ν)<h_{top}(X)$, there exists a shift-invariant measure $μ$ on $X$ such that the systems $(Y,ν,G)$ and $(X,μ,G)$ are isomorphic. In particular, any $K$-shift (consisting of the indicator functions of all maximal $K$-separated sets) containing a free element is universal.

For each of the following conditions, we characterize the pseudovarieties of semigroups V that satisfy it: (i) every epimorphism to a member of V is onto; (ii) every epimorphism to a finite semigroup with domain a member of V is onto; (iii) for every epimorphism from S to T with S in V and T finite, T is also a member of V.

We present second-order algorithms to approximate the solution of a nonlocal Gray-Scott model that is known to generate interesting spatio-temporal structures such as pulse and stripes solutions. Our algorithms rely on a quadrature method for the spatial discretization and the method of lines using a second-order Adams-Bashforth for the time marching. We focus on studying the impact of the type of boundary constraints, e.g. nonlocal Dirichlet/Neumann or local periodic, and the type of nonlocal diffusion, i.e. integral operator with thin- or fat-tailed kernels, on the generation of pulse solutions. Our numerical investigations show that when the spread of the kernel is large, i.e. when the model is nonlocal, both the type of kernels and type of boundary constraints have a strong impact on the solutions profiles.

We consider a polling system with two queues, where a single server is attending the queues in a cyclic order and requires non-zero switching times to switch between the queues. Our aim is to identify a fairly general and comprehensive class of Markovian switching policies that renders the system stable. Potentially a class of policies that can cover the Pareto frontier related to individual-queue-centric performance measures like the stationary expected number of waiting customers in each queue; for instance, such a class of policies is identified recently for a polling system near the fluid regime (with large arrival and departure rates), and we aim to include that class. We also aim to include a second class that facilitates switching between the queues at the instance the occupancy in the opposite queue crosses a threshold and when that in the visiting queue is below a threshold (this inclusion facilitates design of `robust' polling systems). Towards this, we consider a class of two-phase switching policies, which includes the above mentioned classes. In the maximum generality, our policies can be represented by eight parameters, while two parameters are sufficient to represent the aforementioned classes. We provide simple conditions to identify the sub-class of switching policies that ensure system stability. By numerically tuning the parameters of the proposed class, we illustrate that the proposed class can cover the Pareto frontier for the stationary expected number of customers in the two queues.

Planes are familiar mathematical objects which lie at the subtle boundary between continuous geometry and discrete combinatorics. A plane is geometrical, certainly, but the ways that two planes can interact break cleanly into discrete sets: the planes can intersect or not. Here we review how oriented matroids can be used to try to capture the combinatorial aspect, giving a way to encode with finite sets all the ways that $n$ planes can interact. We mention how the one-to-one correspondence breaks down in 2 dimensions for 9 lines, and in 3D for 8 planes. We include illustrations of all the types of plane arrangements using $n=4$ and 5.

Some of the most challenging problems in Timaeus' cosmology arise from the geometry of a universe without any void. On the one hand, the universe is spherical in shape; on the other hand, it must be entirely filled with the four basic particles that make up all bodies in the universe, each shaped like one of four regular polyhedra (cubes, tetrahedra, octahedra and icosahedra). The faces of all these particles are composed of right triangles. However, this leads to two mathematical impossibilities. 1. Obtaining a spherical surface from linear surfaces, as it is impossible to create a circle from straight lines. 2. Obtaining a complete tiling of a sphere using regular polyhedra, without any voids or intersections between these polyhedra. The first problem is addressed in another article slated for publication. In the present one, our focus will be on the second problem, for which we will present a solution within the framework of Timaeus' cosmology. The crux of this solution lies in a feature of Timaeus' universe that sets it apart from almost all ancient cosmologies. Instead of being composed of rigid parts, it is a dynamic living body in which all basic components are in constant motion, continuously undergoing both destruction and reconstruction. In the first part, we examine the main features of Timaeus' cosmology relevant to our issue. In the second part, we analyze the paradox in details and its consequences for Timaeus' cosmology. We then discuss the common 'solutions' and highlight their shortcomings. Finally, we propose a solution that we believe is consistent with Plato's text and independent of the choice of the major schools of interpretation of the Timaeus.

Let $\mathcal{L}$ be the closure of the set of all real numbers $α$, such that there exist infinitely many integers $n$, such that $α=\log\frac{d(n+1)}{d(n)}$, where $d$ is the number of divisors of $n$. We give improved lower bounds for the density of $\mathcal{L}$.

We prove a new bound to the exponential sum of the form $$ \sum_{h \sim H}δ_h \mathop{\sum_{m\sim M}\sum_{n\sim N}}_{mn\sim x}a_{m}b_{n}\e\big(αmn + h(mn + u)^γ\big), $$ by a new approach to the Type I sum. The sum can be applied to many problems related to Piatetski-Shapiro primes, which are primes of the form $\lfloor n^c \rfloor$. In this paper, we improve the admissible range of the Balog-Friedlander condition, which leads to an improvement to the ternary Goldbach problem with Piatetski-Shapiro primes. We also investigate the distribution of Piatetski-Shapiro primes in arithmetic progressions, Piatetski-Shapiro primes in the intersection of multiple Beatty sequences and so on.

This article contains a new result in Fourier analysis concerning jump type discontinuities.

We prove that the subgroup graph of a finite group $G$ is regular if and only if $G$ is cyclic with square-free order.

In this paper, we will give a new proof for a known result of the mean square of Riemann zeta-function.

We consider a bounded open subset $Ω$ of ${\mathbb{R}}^n$ of class $C^{1,α}$ for some $α\in]0,1[$, and we define a distributional outward unit normal derivative for $α$-Hölder continuous solutions of the Helmholtz equation in the exterior of $Ω$ that may not have a classical outward unit normal derivative at the boundary points of $Ω$ and that may have an infinite Dirichlet integral around the boundary of $Ω$. Namely for solutions that do not belong to the classical variational setting. Then we show a Schauder boundary regularity result for $α$-Hölder continuous functions that have the Laplace operator in a Schauder space of negative exponent and we prove a uniqueness theorem for $α$-Hölder continuous solutions of the exterior Dirichlet and impedance boundary value problems for the Helmholtz equation that satisfy the Sommerfeld radiation condition at infinity in the above mentioned nonvariational setting.

This monograph presents a comprehensive analysis of the technological foundations of management decision-making in the reconstruction of complex gas pipeline systems. The study addresses the challenges posed by the aging infrastructure of gas supply networks and explores advanced strategies to improve their reliability, efficiency, and automation. Particular attention is given to the reconstruction of pipelines with various configurations linear, looped, and parallel systems under non-stationary gas flow conditions. The proposed models and methodologies offer solutions for optimizing operational parameters, improving emergency valve response, and ensuring uninterrupted gas supply through advanced management systems and data-driven decision support tools. Emphasis is placed on the integration of modern technologies, system theory, and feedback mechanisms in the design and operation of reconstructed pipeline systems. This work is intended for engineers, system designers, and researchers in the fields of gas supply, systems engineering, and energy infrastructure.

In this paper, some Jensen's type inequalities between quaternionic bounded selfadjoint operators on quaternionic Hilbert spaces are proved, using a log-convex function. Also, by applying a specific log-convex function, some particular cases of operator inequalities are obtained.